In many materials processing routes, objects such as bubbles or particles interact with a moving solidification front. The output of this confrontation, from instantaneous encapsulation to complete rejection of objects, regulates the solidified microstructure and the spatial distribution of the objects, and thus the final properties of the materials. Here we investigate by in situ cryo-confocal microscopy how thermal conductivity and solute compete to control the interfacial curvature of the solidification front. We first validate the prediction of physical models in absence of solute, and then demonstrate the dominating effect of solute. In the case of cellular front morphology, we show that thermal effects depending on the particle conductivity induce a distortion of ice crystals thereby modifying the final microstructure of the solid. Overall our results show that to successfully predict and control solidification microstructure in the presence of objects, physical models that take into account both thermal and long-range solute effects are now required.